Supersymmetric solutions of SU(2)-Fayet-Iliopoulos-gauged N=2,d=4 supergravity

We explore the construction of supersymmetric solutions of theories of N=2,d=4 supergravity with a SU(2) gauging and SU(2) Fayet-Iliopoulos terms. In these theories an SU(2) isometry subgroup of the Special Kahler manifold is gauged together with a SU(2) R-symmetry subgroup. We construct several solutions of the CP3 quadratic model directly in four dimensions and of the ST[2,6] model by dimensional reduction of the solutions found by Cariglia and Mac Conamhna in N=(1,0),d=6 supergravity with the same kind of gauging. In the CP3 model, we construct an AdS2xS2 solution which is only 1/8 BPS and an RxH3 solution that also preserves 1 of the 8 possible supersymmetries. We show how to use dimensional reduction as in the ungauged case to obtain RnxSm and also AdSnxSm-type solutions (with different radii) in - and 4 dimensions from the 6-dimensional AdS3xS3 solution.


Introduction
The study of supersymmetric solution of supergravity theories has been one of the most fruitful areas of research in this field over the last few years providing, for instance, backgrounds for string theory with clear spacetime interpretation such as black holes, rings, or branes, their near-horizon geometries, pp-waves etc. on which the strings can be quantized consistently. Thus, these solutions have provided the earliest connections between gravity solutions and 2-dimensional conformal field theories (the superstring worldsheet theories) whose states can be counted using standard techniques, paving the way for more general correspondences.
The supersymmetric solutions of many (classes of) supergravity theories have been classified/characterized by now, and, therefore, the independent variables that enter in their fields and the equations that they must obey are well known. However, the explicit construction of these solutions can still be a difficult problem when the equations that need to be solved are non-linear as it is often the case in gauged supergravities, specially with non-Abelian Yang-Mills fields. In this paper we are going to deal with this problem in the context of N = 2, d = 4 gauged supergravities. N = 2, d = 4 supergravities admit several kinds of gaugings: 1 1. One can just gauge a non-Abelian subgroup of the isometry group of the Special Kähler manifold of the complex scalars from the vector multiplets. 2 This is the simplest possibility: it does not involve the hypermultiplets and trying to gauge an Abelian isometry only would have no effect since all the terms that would have to be added (proportional, for instance, to the Killing vector) vanish identically. In absence of hypermultiplets, these theories have been called in Refs. [4,5] N = 2, d = 4 Super-Einstein-Yang-Mills (SEYM) because they are the simplest N = 2 supersymmetrization of the Einstein-Yang-Mills theories.

2.
One can gauge a general subgroup of the isometry group of the Quaternionic Kähler manifold of the scalars in the hypermultiplets. 3 Since this requires coupling to a set of gauge vector fields transforming in the adjoint of the gauge group and the available vectors come in supermultiplets that also contain scalars in a Special Kähler manifold, the gauge group must also be a subgroup of the isometry group of the Special Kähler manifold and must necessarily act on the hypermultiplets and vector multiplets simultaneously. It must act in the adjoint representation on the latter. This case can be considered an extension of the previous one in which the hypermultiplets are not mere spectators anymore. There is, however, a very important difference: Abelian gaugings are non-trivial in this setting in the Quaternionic Kähler sector.
Furthermore, there are two particular sub-cases to be considered: (a) When the gauge group contains a U(1) factor, it is possible to deform the gauged theory adding a constant Fayet-Iliopoulos (FI) term. Remarkably enough, this term survives when the hypermultiplets are truncated and, in that case one obtains a non-trivial U(1)-FI-gauged N = 2, d = 4 supergravity with only vector multiplets. There is no contradiction with the impossibility of gauging Abelian isometries of the Special Kähler manifold stated above because, in this case, the global symmetry being gauged is a U(1) subgroup of the SU(2) factor of the R-symmetry group U(2) =U(1)×SU (2). 4 (b) When the gauge group contains a SU(2) factor, again it is possible to deform the gauged theory adding a constant SU(2) FI term. Again, somewhat surprisingly, this term survives the truncation of the hypermultiplets that gives a SU(2)-FI-gauged N = 2, d = 4 with only vector multiplets. These theories can be seen as deformations of the N = 2, d = 4 SEYM theories in which the SU(2) factor of the R-symmetry group is gauged simultaneously with an SU(2) subgroup of the isometry group of the Special Kähler manifold.
Our goal in this paper is to search for timelike supersymmetric solutions of this last class of gauged supergravities: SU(2)-FI-gauged N = 2, d = 4 supergravities with no hypermultiplets.
Many solutions of the ungauged, U(1)-FI-gauged and SU(2) SEYM theories have been constructed in the literature but, so far, no supersymmetric solution of SU(2)-FI-gauged theories is explicity known. This is due to the complexity of the theories and of the equations that need to be solved to construct supersymmetric solutions. Therefore, our very first task will be to describe carefully the structure of SU(2)-FIgauged N = 2, d = 4 supergravities with no hypermultiplets (Section 1) and the second will be to spell out in detail the characterization of the timelike supersymmetric solutions of these theories found in Ref. [6] (Section 2), showing that, according to the results of Ref. [14], none of them will be maximally supersymmetric. We will, then (Section 3), consider the simplest theory that admits an SU(2) gauging, the so-called CP 3 model, and we will perform the gauging with FI terms, constructing explicitly the scalar potential.
In Section 4 we setup and try to solve by using different methods and ansatzs the equations that the elementary building blocks of supersymmetric solutions must satisfy in the CP 3 model. We present 3 different solutions. Finally, in Section 5 we try a different approach which is only valid for ST [2, n] models with n ≥ 6: the authors of Ref. [15] constructed several timelike supersymmetric solutions of an SU(2)-FI-gauged N = (1, 0), d = 6 theory and, by dimensional reduction, we can obtain solutions of the corresponding SU(2)-FI-gauged N = 1, d = 5 and N = 2, d = 4 theories. 5 Unfortunately, most of the solutions we obtain in this way do not have an good asymptotically behavior (flat, AdS,..) nor they are in general free of naked singularities. The exception is the AdS 3 ×S 2 solution one which can be obtained from the AdS 3 ×S 3 one in 6 dimensions. There are other possibilities to obtain solutions of the same type in 5 and 4 dimensions that we explain in detail. Section 6 contains our conclusions and directions for future work.

SU(2)-FI-gauged
In this section we are going to review quickly the kind of theories we will be dealing with. For more details, the reader is referred to Refs. [1,6,3], whose conventions we follow here. More information on the construction of these theories can be found in Ref. [2]. We are considering theories of N = 2, d = 4 supergravity, where the supergravity multiplet contains the metric g µν and the graviphoton vector field A 0 µ plus two gravitini ψ I µ , I, J, . . . = 1, 2, coupled to n vector multiplets, each of them consisting of a complex scalar Z i and a vector field A i µ plus two gaugini λ i I , i = 1, · · · , n. All the vector fields are combined into A Λ µ , Λ, Σ, . . . = 0, 1, · · · , n. The complex scalar parametrize a Special-Kähler manifold. The Special-Kähler structure, which determines the Kähler potential K (and, hence, the Kähler metric G ij * = ∂ i ∂ j * K of the scalar σ-model) and the period matrix N ΛΣ (Z, Z * ) that describes the coupling of the scalars to the vector field strengths (kinetic matrices), is completely determined by the canonical covariantly-holomorphic symplectic section 6 V = L Λ M Λ or by a prepotential F . These two objects determine completely the ungauged theory.
The global symmetries of a theory of N = 2, d = 4 supergravity coupled to vector supermultiplets are the holomorphic isometries of the Kähler metric that also preserve the rest of the Special-Kähler structure 7 and the R-symmetry group U(2) which only 5 The solutions of SU(2)-FI-gauged N = 1, d = 5 have not been received much attention, either, and, to the best of our knowledge, none have been presented in the literature up to this moment. 6 We will also use acts on the indices I, J, K of the fermion fields in the fundamental representation. When the group of isometries that are also global symmetries of the theory includes a non-Abelian subgroup 8 which acts in the adjoint representation on a subset of the vector supermultiplets, one can gauge it: if the holomorphic isometries are global symmetries of the theory, there are holomorphic Killing vectors k Λ (Z) and associated symplectic generators of the gauge group T Λ satisfying the same Lie algebra where the f ΛΣ Ω are the structure constants. 9 To gauge the theory, the scalar and vector field strengths are modified in the standard way to make them covariant under the local transformations: 10 Here g is the gauge coupling constant. Furthermore, supersymmetry requires the addition of a scalar potential which turns out to be non-negative. The result is the minimal N = 2 supersymmetrization of the bosonic Einstein-Yang-Mills theory for that gauge group. These theories were called N = 2 Super-Einstein-Yang-Mills (SEYM) theories and their timelike supersymmetric solutions were characterized in Ref. [5] and studied in Refs. [4,16,17,18]. Gauging a subgroup of the R-symmetry group seems to be a different choice, and, indeed it is if the subgroup is Abelian (U(1) ⊂SU(2) is the only possibility), because, as we mentioned above, Abelian holomorphic isometries cannot be gauged in these theories. The gauging is done via Fayet-Iliopoulos (FI) terms. The supersymmetric solutions of these theories have been classified and studied in Refs. [10,12].
However, when this subgroup is non-Abelian (SU(2) is the only possibility, via FI terms as well) it turns out that choice is not so different, actually: to gauge it we need gauge vector fields transforming in the adjoint representation of the gauge group. This implies that the whole supermultiplets, and, in particular the complex scalars, must transform in the adjoint representation leaving the whole Special-Kähler structure (and, in particular, the Kähler metric) invariant. Thus, if one gauges a SU(2) subgroup of the R-symmetry group one has to gauge at the same time a SU(2) isometry subgroup of the global symmetry group and one can see the resulting theory as a deformation, via FI terms, of a N = 2 SEYM theory with a gauge group that includes a SU(2) factor so that, for a subset of the vector indices Λ, Σ, . . . that we are going to denote by the indices x, y, . . ., that only take 3 possible values, the structure constants are those of SU(2): These are the theories we are interested in. Their timelike supersymmetric solutions were classified as part of the general case studied in Ref. [6]. In the examples we will consider there will be no other factors in the gauge group apart from the SU(2) one.
Since the difference between these theories and the N = 2 SEYM theories is the action of the gauge group on the fermions, at the bosonic level the only difference one sees is the scalar potential, which contains additional terms and is no longer nonnegative. The scalar and vector field strengths still take the form Eqs. (1.3) and (1.4). The bosonic action is given by where the scalar potential V(Z, Z * ) is given by where the objects f Λ i are the upper components of the Kähler-covariant derivatives of the canonical symplectic section , P Λ are the holomorphic momentum maps, and the triholomorphic momentum maps P Λ x , x, y, . . . = 1, 2, 3, are assumed to be of the form for ξ = 0, 1 11 and constant tensors e Λ x nonzero for Λ in the range of the SU(2) factor satisfying 9) or, taking into account Eq. (1.5), 11 The role of this unphysical parameter will be to help us set to zero the FI terms, recovering the N = 2, d = 4 SEYM theories. ε xy z e y y e z z = −ε xyz e z x . (1.10) With no loss of generality we will choose the simplest solution These constant triholomorphic momentum maps give rise to SU(2) FI terms and often we will use that name for them. With this choice, the scalar potential takes the simple form Observe that the first term may contain the contribution of other (necessarily non-Abelian) gauge factors apart from the SU(2) one labeled by x, y, . . . In the examples that we are going to consider we will not include that possibility and, therefore, the sum over indices Λ, Σ, . . . will be restricted to a sum over the SU(2) indices x, y, . . . There are other differences between these theories and the SEYM ones in the covariant derivatives of all the fermions (which now transform linearly under the gauge group in the I, J, . . . indices) 12 and in the supersymmetry transformations as well. We will not deal directly with them and, therefore, we will not describe them here, for the sake of simplicity. All this information can be found in the references mentioned at the beginning of this section.

Timelike supersymmetric solutions
The timelike supersymmetric solutions of the theories introduced in the previous section have been characterized in Ref. [6], where the most general gauging of these theories was considered. In this section we are going to particularize the results obtained there to the case of the theories we are dealing with, with only SU(2) as gauge group and with the choice of FI terms Eqs. (1.8) and (1.11).
In order to describe the form of these solutions we start by introducing an auxiliary object X with the same Kähler weight as the canonical symplectic section V M so that the quotient V M /X has vanishing Kähler weight. Then, we define two real symplectic vectors R M , I M For any model of N = 2, d = 4 supergravity (or, equivalently, for any canonical symplectic section V M ) the components R M can, in principle, be expressed entirely in terms of the components I M , although, in practice, this can be very hard to do for certain models. This is often referred to as "solving the stabilization equations" or as "solving the Freudenthal duality equations". We will assume that this has been done and, indeed, that will be the case in the models we will study here. Then, the symplectic product R M I M = R | I = R Λ I Λ − R Λ I Λ is a function of the I M only that we call the Hesse potential Now we are ready to describe the form of the fields of the timelike supersymmetric solutions: 1. First of all, their metric can always be written in the conformastationary form 13 3) The elements that enter in this expression are required to have a specific form or satisfy certain equations: (a) The metric function e −2U is given by the Hesse potential and these must satisfy the equation (2.8) 13 We use hats to denote differential forms.
(c) The 1-formω satisfies the equation (in tangent 3-dimensional space) whereD is the covariant derivative w.r.t. the effective 3-dimensional gauge connection: 12) and the space components A Λ x together with the functions I M are determined by the following generalization of the Bogomol'nyi equations written again in tangent 3-dimensional space:

The time-component of the vector fields has been gauge-fixed to
where we have defined 14 3. Finally, the scalars are given by There are no dual 1-forms A Λ in this formulation of the gauged theory.

Maximally supersymmetric vacua
Before we start looking for explicit examples of supersymmetric solutions, it is worth discussing the possible existence of maximally supersymmetric solutions. According to the results of Ref. [14] the supersymmetric solutions of these theories, if any, must be of the same kind as those of the corresponding ungauged theories: in absence of electromagnetic fluxes, Minkowski spacetime M 4 or anti-de Sitter spacetime AdS 4 and, in presence of fluxes, Bertotti-Robinson spacetimes AdS 2 ×S 2 [19,20] or Kowalski-Glikman homogeneous pp-wave spacetimes KG 4 [21]. Furthermore, maximally supersymmetric solutions in gauged supergravities are characterized by the vanishing of all the fermion shifts and of the R-symmetry connection [14]. For the N = 2, d = 4 the different possibilities were analyzed in detail in Ref. [22]. The maximally supersymmetric solutions with zero curvature (M 4 , AdS 2 ×S 2 and KG 4 ) must have identically vanishing triholomorphic momentum maps P Λ x = 0, which is not possible in the case we are considering. The remaining possibility is the only maximally supersymmetric solution with negative curvature: AdS 4 . the following conditions have to be satisfied in this case: With our choice of FI terms (1.8),(1.11) these conditions take the form Using the choice of coordinates Z i = X i /X 0 and the gauge X 0 = 1, it is not difficult to see, from the definition f Λ i = e K 2 D i X Λ that it is not possible to satisfy all the Eqs. (2.23) at the same time.
We conclude that these theories do not admit maximally supersymmetric vacua.

The SU(2) gauging of the CP model
In order to search for explicit examples of supersymmetric solutions we must specify the model of N = 2, d = 4 supergravity we work with. The simplest example that admits an SU(2) gauging is the CP 3 model. Here we quickly review it. This model has 3 vector multiplets and the quadratic prepotential We can define the 3 complex scalars, which parametrize a U(1, 3)/(U(1)×U (3)) coset space, by The Kähler potential, the Kähler metric (which is the standard Bergman metric for the symmetric space U(1, 3)/(U(1)×U (3)) [23]) and its inverse in the X 0 = 1 gauge are given by 4) which implies that the complex scalars are constrained to the region The covariantly holomorphic symplectic section V M , its Kähler-covariant derivative U i = D i V and the period matrix are given by (3.6) For later use we also quote and the Hesse potential Since the scalars parametrize the symmetric space U(1, 3)/(U(1)×U (3)), the metric (and, indeed, the whole model) is invariant under global U(1, 3) = U(1) × SU(1, 3) transformations. We are interested in the SU(1, 3) subgroup whose SO(3) subgroup we are going to gauge.
The special coordinates X Λ transform in the fundamental representation of SU(1, 3): and, according to their definition, the complex scalars transform non-linearly, as We will use the metric η ΛΓ and its inverse to lower and raise the indices of the SU (1, 3) transformations Λ Λ Σ . These transformations leave the Kähler potential invariant up to Kähler transfor- which implies the exact invariance of the Kähler metric. The symplectic section V N is also left invariant by the combined action of the symplectic transformation that gives the embedding of the group SU(1, 3) in the symplectic group Sp(8, R) and a Kähler transformation with the parameter f (Z) given in Eq. (3.11). This proves the invariance of the whole model of N = 2, d = 4 supergravity.
are traceless and such that T m ΛΣ ≡ η ΛΓ T m Γ Σ is anti-Hermitian. Then, the corresponding sp(1, 3) generators, whose exponentiation gives the matrix Eq. The holomorphic Killing vectors that generate the transformations of the scalars Eqs. (3.10) can be written in the form The holomorphic functions λ m (Z) defined through are given by 18) and the holomorphic momentum maps P m , defined through the relation are given by The SU(2) subgroup that we are going to gauge acts in the adjoint representation on the special coordinates X i and on the physical scalars Z i , leaving exactly invariant X 0 , the prepotential and the Kähler potential (so f = λ = 0). We are going to use the indices x, y, z, · · · = 1, 2, 3 to denote the scalars of the gauged directions, instead of i, j, · · · . Thus, the vector fields A Λ split into A 0 and A x , the physical scalars are Z x , the non-vanishing structure constants and the generators are 15 and the holomorphic momentum maps and Killing vectors are given by where α is the angle between the 3-vectors eZ x and mZ y . Observe that the first term in the potential is non-negative but also bounded above due to Eq. (3.5): but the second, which is associated to the FI terms, is unbounded below (e K ∈ (1, ∞)): We have explored the minima of this potential and we have found that there is a minimum when all the scalar fields vanish, when one of them vanishes, when two of them are equal or when two of them are real, but the potential is not negative for any of these minima and, therefore, we have not been able to find any (necessarily non-maximally supersymmetric) AdS 4 vacuum in this theory.
As we have already mentioned, the choice of this specific model is due to its simplicity; in particular, its Freudenthal duality equations can easily be solved: (so that R Λ = 0) in order to simplify the equations. In particular, with this choice, the form ω is closed, and we set it to zero. The equations that remain to be solved are For ξ = 1, D ξV x = DV x and for ξ = 0, (when the FI terms vanish) D ξV x = dV x and the last equation would be solved by choosing coordinatesV x = dx x .
The integrability condition of the last equation can be obtained by acting with D on both sides and using the Ricci identity (ξ = 0) We find, up to the overall factor gξ which is satisfied if Eq. (4.3) holds.

Hedgehog Ansatz
it is natural to start by looking for spherically-symmetric solutions. We can adopt the hedgehog Ansatz for the gauge field A x m and the corresponding "Higgs field" Φ x : 16 We can also assume that the 3-dimensional metric γ mn is conformally flat and choose Dreibeins of the form We can also safely assume that The Ansatz for the Abelian vector field A 0 m cannot be spherically symmetric: we know that the potential of the Dirac monopole is not spherically symmetric even though the field strength is. If the unit vector s m indicates the direction of the Dirac string, the Dirac monopole potential can be written in the form We can make the following Ansatz in this case: so the function k can have additional dependence on r (not through w). Substituting this Ansatz into Eqs. (4.2)-(4.4) we get the following differential equations: x m ∂ m k = 0 , (4.16) where primes indicate differentiation with respect to r, which is the only argument of the functions Φ 0 , f , h, V. Eq. (4.16) above implies that k is a function of w only and we are left with and This is the only term in Eq. (4.17) that depends on s m and that dependence must disappear because the corresponding equation is spherically symmetric. Therefore, we must require that for some constant C. This equation can be integrated to give for some other integration constant D. The standard form of the Dirac monopole is recovered when we choose C = D = p/2. Then, Eq. (4.17) becomes and we are left with a non-autonomous system of 4 ordinary differential equations for 4 variables f , h, V, Φ 0 that generalizes Protogenov's [24]. The next step is to try to rewrite this system as an autonomous system by a change of variables. For the Protogenov system this is explained in Ref. [16]. Actually, the same change of variables works here. Defining and combining the differential equations we arrive at the autonomous system When ξ = 0, the third equation is solved by V = constant and, setting that constant to 1, the first two equations become those of the Protogenov system and involve only two variables: N and I. When ξ = 1 the four equations are coupled in a non-trivial way and we have to make additional assumptions in order to simplify the system and find solutions.
Observe that there are no solutions with vanishing scalars, that is, with I = 0: setting I = 0 in Eqs. (4.24) and (4.25) and combining them to eliminate K we obtain a differential equation that only involves N and can be integrated to give N = − tanh η + α where α is some integration constant. Then, Eq. (4.25) cannot be satisfied for any real V or K.
A further change of variables, I = V I and K = VK, allows us to rewrite the system in a simpler way: This system admits a solution in which N, I and K are constants: the first three equations are algebraic and the fourth is trivial to solve). This allows us to obtain the first solution of this theory.

Solution 1: AdS 2 × S 2
With no loss of generality we can assume I to be positive, and the solution, dependent on two constants I, v is given by: (4.32) The physical fields are then given by: This metric turns out to be that of AdS 2 × S 2 (with different radii), independently of the value of I, as can be seen performing the following change of variables, (4.34) which leads to: The potential (3.26) assumes in this situation a constant value, which can be negative for certain values of the parameter I: By construction this solution is supersymmetric. In order to determine which fraction of the total supersymmetry it preserves (the minimal amount is 1 8 ) we take advantage of the analysis performed in Ref. [6]: the gaugini Killing Spinor Equation is solved imposing three projection operators, each of which projects out half of the components of the Killing spinor. However, if some gaugini's shifts vanish identically for the configuration we are examining, the corresponding projector does not need to be imposed, and the supersymmetry preserved can be larger. From Eqs. (3.4) and (3.6) we get, for the model we are dealing with,

Another Ansatz
In order to generalize of the Ansatz we made in Section 4.1 we are going to relax Eq. (4.9): it will have the same form but now we will allow V to be an arbitrary (that is: not necessarily sphericallysymmetric) function of the coordinates x m . With this choice, Eq. (4.4) can be solved by for some triplet of arbitrary functions h m that, in particular, can vanish identically. We consider first this possibility.

Solution 2
Let us consider the Ansatz (4.40),(4.41) making some further assumptions: h m = 0 and all the functions involved depend on a single direction, say x 1 , so that This Ansatz is adequate to find domain-wall-type solutions.
Under these assumptions, Eq. (4.2) implies that the only non-trivial component of F 0 mn is F 0 23 . However, since, by assumption, the components A 0 2,3 are functions of x 1 only, they must be constants and the purely spatial components of the field strength F 0 mn must vanish identically.
The equations in (4.2) and (4.3) that remain to be solved are and can be rewritten in this form that can be immediately integrated, giving where p 0 , p 1 and v are integration constants. The metric function for these solutions is e −2U = (Φ 0 ) 2 − (Φ 1 ) 2 = −p 1 V(x 1 ) and the complete metric has the form We must set p 0 = 0 because, otherwise, Φ 0 = 0 and the metric function would always be negative and we must require p 1 V < 0 so e −2U > 0. The profile of e −2U changes dramatically with the integration constants and it is not easy to find physically meaningful solutions. One of the few simple examples that we have found corresponds to the choice, p 0 = −1, for which e −2U (x 1 ) is positive in an interval of the real line (see the figure where we have represented the inverse, e 2U ).
At the boundary of that region e −2U and V blow up, and so does the scalar potential, which in this case is given by On the other hand, the condition W ix = 0 cannot be satisfied for any x, meaning that the solution is 1 8 -BPS.

Solution 3
If, in the context of the Ansatz Eqs. (4.40),(4.41), we still assume that all the functions involved depend only on x 1 but we do not assume the vanishing of h m , the non-trivial components of Eq. (4.4) take the form those of Eq. (4.3) take the form and, finally, those of Eq. (4.2) take the form 60) It is immediate to conclude that and the equations that remain to be solved are This system of equations can be simplified by setting Φ 1 = 0; in this way, the resulting equations are easy to solve, and the solution is determined by the following non-vanishing fields: where b is an integration constant. The spatial part of the metric is the metric of a 3-dimensional hyperboloid in coordinates analogous to the Poincaré coordinates of AdS 3 17 and, therefore, the complete 17 If we define the hyperboloid as the hypersurface metric has the geometry of R × H 3 and it is supported only by a non-Abelian field whose field strength is related to the volume form of H 3 by Usually, p-form field strengths support pof (d − p)-dimensional symmetric spaces. For instance, 2-form field strengths support AdS 2 ×S 2 solutions in 4 dimensions and AdS 2 ×S 3 or AdS 3 ×S 2 solutions in 5 dimensions. In this sense, this solution is exceptional and the exceptionality is related to the rank of the form and to the dimension of the gauge group. The potential is again equal to a positive constant when this configuration is considered, and the amount of supersymmetry preserved by the solution is 1 8 .

Solutions from dimensional reduction
An alternative procedure to construct solutions of a given theory is by dimensional reduction or oxidation of known solutions, provided that there are theories related to the one we are interested in by these mechanisms and that there are known solutions of them which, if they are to be dimensionally reduced, have enough isometries. N = 2, d = 4 supergravity theories are directly related by dimensional reduction or oxidation to other supergravity theories with 8 supercharges. 18 These only exist in d ≤ 6 and, to the best of our knowledge, theories with SU(2) FI gaugings have only been studied in N = (1, 0), d = 6 supergravity coupled to one tensor multiplet and a triplet of vector multiplets in Ref. [15]. This theory is unique 19 and describes a truncation of the Heterotic String compactified on T 4 that includes the metricgμν, a complete 20 (Kalb-Ramond) 2-formBμν, a real scalar (dilaton)φ and the three vector in the R 4 endowed with the metric then, if we parametrize it with coordinates x 1 , x 2 , x 3 the induced metric is (4.76) 18 The relation with theories with different number of supercharges must necessarily involve truncations and constraints on the solutions and we will not consider them here. 19 As different from d = 4, 5 supergravities with 8 supercharges, in the d = 6 case, there is only one model for each possible matter content. 20 That is: not subject to any self-or anti-self-duality (chirality) constraints because it is, actually the sum of the 2-form of the supergravity multiplet and the 2-form of the tensor multiplet, which have opposite chiralities fieldsÃ Ã µ , A = 1, 2, 3. The FI terms induces a simple potential for the dilaton, and the action takes the form [15,26] where g 6 is the 6-dimensional coupling constant. The dimensional reduction of N = (1, 0), d = 6 supergravity theories coupled to tensor and vector multiplets on a circle has been studied and the models of N = 1, d = 5 supergravity coupled to vector multiplets they give rise to have been determined in Ref. [26]. We can use the results in that paper to dimensionally reduce the 6dimensional solutions found in Ref. [15] to solutions of SU(2) FI-gauged N = 1, d = 5 supergravity since the relation between the 6-and 5-dimensional fields of the gauged theories is the same as in the ungauged case, as long as the gauge groups are the same in both theories. These relations are given Appendix A. The 5-dimensional model obtained in the dimensional reduction is completely characterized by the symmetric tensor C 0rs = 1 3! η rs , r, s = 1, . . . , 5. The bosonic fields in this theory are the metricĝμν, the 6 gauge fieldsÂ Iμ , I = 0, · · · , 5, 5 of which,Â rμ , correspond to 5 vector multiplets 21 , and 5 scalar fields. Due to the reduction procedure,Â 0,1,2μ are Abelian fields, whileÂ A+2μ are the three SU (2)  A convenient parametrization isφ r =ĥ r soĥ 0 = 2/(φηφ) ≡φ 0 , where φηφ ≡ φ r η rsφ s . In this parametrization, the last 3 scalarsφ A+2 transform in the adjoint representation of SU(2) and the action of the theory can be written in the compact form whereDμφ 0,1,2 = ∂μφ 0,1,2 ,Dμφ A+2 = ∂μφ A+2 − g 5
(5.5) 21 The reduction of the KR 2-form gives just 2 vector fields.
Observe that the 6-and 5-dimensional gauge coupling constants are related by The dimensional reduction of N = 1, d = 5 supergravity on a circle gives cubic models of N = 2, d = 4 supergravity. Therefore, the CP 3 model cannot be obtained in this way. The model that actually arises in the dimensional reduction of the above 5-dimensional model is the ST [2,6] model, which is characterized by the prepotential 22 where i = 1, 2 · · · , 6 labels the 6 vector multiplets and where the fully symmetric tensor d ijk has as only non-vanishing components d 1αβ = η αβ , where (η αβ ) = diag(+ − · · · −) , and α, β = 2, · · · , 6 . (5.8) The 6 complex scalars parametrize the coset space (5) , (5.9) and the group SO(3) acts in the adjoint on the coordinates α = 4, 5, 6 that we are going to denote with A, B, . . . indices. These are the directions which are gauged. With our conventions, the SL(2,R) SO (2) factor is parametrized by the scalar Z 1 which is often called the axidilaton field since its real and imaginary parts are, respectively, an axion and a dilaton field.
The action of the ST[2, 6] model can be constructed using the standard formulae valid for any cubic model. 23 It has a complicated form that we are not going to use directly and, therefore, we refrain from writing it here. The computation of the scalar potential using the general formula Eq. (1.7) requires the computation of the momentum maps etc., but we can also obtain it by dimensional reduction using the relation between 5-and 4-dimensional fields that can be found, for instance, in Ref. [18]. It takes the extremely simple form (that is: proportional to the exponential of the dilaton field and, therefore, negative definite) where now the 5-and 4-coupling constants are related by More details on this theory and, in particular, on its relation with the toroidal compactification of the Heterotic string can be found in Refs. [17,18]. 23 See, for instance, Ref. [3].
Thus, to summarize this discussion, we can obtain supersymmetric solutions of the above SU(2) FI-gauged supergravities by dimensional reduction of the 6-dimensional supersymmetric solutions constructed in Ref. [15], using the relations in the Appendix. In the rest of this section we are going to do just that for some of those 6-dimensional solutions.

Solution 1
The first solution of Ref. [15] that we are going to reduce to 4 dimensions is given in Section 6.2.1 of that reference and it is, perhaps, the simplest: it is a generalization of the solution with geometry M 4 ×S 2 found by Salam in Sezgin in Ref. [27] that has M 3 ×S 3 metric, a constant dilaton field whose value is proportional to the square of the radius of the S 3 and to the square of the coupling constant, a meronic gauge field and vanishing 2-form. The non-vanishing field are given by , where the σ A are the left-invariant Maurer-Cartan 1-forms satisfying dσ A = 1 2 ε A BC σ B ∧ σ C , dΩ 2 (3) = 1 4 σ A σ A and a is a constant parameter. Reducing along the z coordinate using Eqs. (A.1), we get a solution of the 5dimensional theory with the following non-vanishing fields: This solution belongs to the same class as its 6-dimensional parent: it has constant scalars and a meronic gauge field that support an M 2 ×S 3 geometry. radii. Since this limit is equivalent to setting Q 1 = 0, the AdS 3 ×S 3 near-horizon limit is a supersymmetric solution as well.
If we reduce along the z direction, the following 5-dimensional solution is obtained This solution is singular at r = 0 and it is not asymptotically AdS (or some other known vacuum solution). If we reduce it again along the coordinate φ, defined by dΩ 2 (3) = 1 4 (dφ + cos θ dψ) 2 + dθ 2 + sin 2 θ dψ 2 , we get a 4-dimensional solution which we will refrain from writing explicitly because it has the same problems as the 5dimensional one.
Of course, we could have used this coordinate φ in the reduction from 6 to 5 dimensions. Doing that we get a 5-dimensional solution with the properties similar to those of the 6-dimensional one: where ρ ≡ Q 1/2 2 /r but, again, it is not asymptotically AdS. The r → 0 limit of the complete solution coincides with the solution that one gets by setting Q 1 = 0. Thus, there is a globally regular AdS 3 × S 2 solution in this theory. It could have been obtained directly by dimensional reduction from the 6-dimensional AdS 3 × S 3 solution.
Further reduction along the z coordinate would lead to the same 4-dimensional solution mentioned above. There is, however, other possibilities inspired in the results of Ref. [28], in which the relation between AdS n × S m vacua of the 4-, 5-and 6-dimensional theories with 8 supercharges was studied. The main observation is that, just as S 3 can be seen as a U(1) fibration over S 2 and one gets that S 2 by dimensional reduction along that fiber 24 , AdS 3 can be seen as a U(1) fibration over AdS 2 and, by dimensional reduction along that fiber one gets AdS 2 . Thus, if instead of using the coordinate z along which the 6-dimensional string lies, one uses the U(1) fiber of the AdS 3 in the AdS 3 × S 3 solution, we would have obtained an AdS 2 × S 3 solution in 5 dimensions and then an AdS 2 × S 2 solution in 4 dimensions.
A more general dimensional reduction is possible: one can rotate the two U(1) fibers of the 6-dimensional solution and dimensionally reduce along one of the rotated fibers. As in the ungauged case studied in Ref. [28] one would get a solution that describes geometry of the near-horizon limit of the BMPV black hole in which the remaining U(1) is non-trivially fibered over AdS 2 ×S 2 . This space is obtained in 4 dimensions after dimensional reduction along the remaining fiber.
The main difference with the ungauged case, apart from the presence of non-trivial SU(2) gauge field, is the difference between the radii of the two factors of these metrics.
Carrying out these alternative dimensional reductions following Ref. [28] is straightforward, albeit quite involved due to the necessity to rewrite the 6-dimensional solution in different coordinates. We leave it for a future publication.

Conclusions
Exploring the space of the supersymmetric solutions of a supergravity theory is one of the most elementary steps one can take to get a more complete understanding of its structure, providing information about the possible vacua and some of the solitonic objects that can exist on it. In this paper we have taken this step for two particular examples (the CP 3 and ST[2, 6] models) of a wide class of theories with a class of gaugings that has been overlooked so far: SU(2)-FI-gauged N = 2, d = 4 supergravity.
Although, as we have shown, no maximally supersymmetric solutions exist in these theories, there are non-maximally-supersymmetric solutions that can be seen as a deformation of the maximally supersymmetric vacua of the ungauged theory, such as the AdS 2 ×S 2 solutions with different radii. Actually, in the ST [2,6] model, the AdS 2 ×S 2 solution must have a higher-dimensional origin analogous to that of the ungauged case [28] and we have indicated the existence of a family of vacua of N = 1, d = 5 supergravity similar to the near-horizon geometry of the 5-dimensional BMPV black-hole originating in the AdS 3 ×S 3 solution of N = (1, 0), d = 6 supergravity with different radii constructed by Cariglia and Mac Conamhna in Ref. [15].
It is likely the existence of deformed versions of the rest of the maximally supersymmetric vacua of N = 1, d = 5 supergravity (Hpp-waves and Gödel spacetimes [29,30]). It may be possible to obtain them from the above-mentioned solutions by different limiting procedures [31]. On the other hand, it would be interesting to find complete black-hole and black-string solutions whose near-horizon geometries were precisely the AdS m ×S n solutions we have discussed, but it is not guaranteed that they are always going to exist and their asymptotic behaviour is uncertain.
Apart from these solutions we have found solutions whose geometry is of the form M m × S n in 4 and 5 dimensions which descend from a 6-dimensional solution of the same kind and a solution of the CP 3 model with R × H 3 geometry which deserves further study. Work in this direction is in progress [31].